University of Helsinki, Faculty of Science, Department of Mathematics and Statistics

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Orponen, Tuomas

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2012-10-05T07:43:11Z

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2012-10-17

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2012-10-05T07:43:11Z

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2012-10-27

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URN:ISBN:978-952-10-8269-6

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http://hdl.handle.net/10138/37098

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The dissertation Exceptional sets in projection and slicing theorems contains a treatment of two classical topics in fractal geometry: projections and slicing. The thesis consists of an introductory chapter and two scientific articles; the new results extend a long line of research originated by J. M. Marstrand in 1954.
The first paper deals with projecting a planar set K onto lines. The fractal geometer is interested in the following question: what is the relation between the dimensions of K and its projections? In 1954, Marstrand proved that if the dimension of K lies between zero and one, then the projections tend to preserve dimension; for almost every line the dimension of the projection equals dim K. During its nearly 60 years of existence, this theme has spawned countless variations. In the thesis, special attention is given to scrutinizing the words almost every line in Marstrand s theorem. The words cannot be entirely omitted (an illustrative example is given by projecting the y-axis onto the x-axis), but they can be sharpened in many cases. The definition used by Marstrand allows for a fairly large set of exceptional lines , the projection onto which fails to preserve the dimension of K. It turns out that better bounds for the size of this exceptional set can be obtained through a more intricate analysis.
The second paper is thematically close akin to the first; it takes up another 1954 result by Marstrand, the slicing theorem , and examines the exceptional set estimates therein. To explain the slicing theorem, fix a planar set K with dimension greater than one. This time, the set K is intersected with various planar lines. What is the dimension of these slices of K? In general, one cannot expect to find a single constant answering the question: if K is bounded, many lines evade K altogether, and the corresponding slices have dimension zero. However, not all slices of K can be so small. Marstrand showed that in almost every direction many lines meet K in a set of dimension dim K 1. In Marstrand s original formulation, the same definition for the words almost every was used as in the projection theorem, and, again, bounds for the size of the exceptional set can be improved with new techniques. In the thesis, similar estimates are also derived in a variant of the theorem where planar lines are replaced by more complicated curves.